ERGODIC PROPERTIES OF THE BOLYAI-RÉNYI EXPANSION

Oliver Jenkinson and Mark Pollicott

Institut de Mathématiques de Luminy and University of Manchester

Abstract. We consider the ergodic properties of the Bolyai-Rényi expansion for real numbers introduced in [Bol], [Ren]. We compute the entropy of the absolutely continuous invariant probability measure μ of the associated dynamical system, resolving a conjecture of Bosma, Dajani and Kraaikamp [BDK]. We also compute the frequencies with which the various digits occur in the expansions of Lebesgue almost every number. Finally, we calculate the rate of decay of correlations with respect to μ.

1. Introduction

In his influential 1957 paper [Ren], Rényi considered the representation of real numbers x ∈ [0, 1] in terms of various “f-expansions” of the form

\begin{aligned}x & = f (a_{1} + f (a_{2} + f (a_{3} + …))) \\ & = lim_{n→∞} f (a_{1} + f (a_{2} + … f (a_{n}) …)) \end{aligned}

for certain integer digits a_{i} ≥ 0. For example, the function f(x) = 1/x gives the familiar continued fraction expansion

x = \frac{1}{a_{1} + _{\frac{1}{a_{2}+ \frac{1}{a_{3}+···}}}} .

The choice f(x) = x/β, for β > 1 gives the β-expansion of a number, which is a natural generalisation of its decimal expansion. Rényi also considered the choice f(x) = \sqrt{1 + x} − 1, leading to the expansion

x = −1 + \sqrt{a_{1} + \sqrt{a_{2} + \sqrt{a_{3} + …}}},

where each a_{i} ∈ {0, 1, 2}.

This expansion had been originally introduced by Bolyai in 1832 (see the 1897 volume [Bol]) to approximate roots of certain equations. It has also recently been considered by Bosma, Dajani & Kraaikamp [BDK].

Typeset by AMS-TEX

We would like to thank Cor Kraaikamp for suggesting this problem to us.

The Bolyai-Rényi expansion of a number x is given (see section 2 for details) by iterating the following map T of the unit interval,

T (x) = \{ \begin{matrix}x^{2} + 2x & if 0 ≤ x < \sqrt{2} − 1 \\ x^{2} + 2x − 1 & if \sqrt{2} − 1 ≤ x < \sqrt{3} − 1 \\ x^{2} + 2x − 2 & if \sqrt{3} − 1 ≤ x ≤ 1, \end{matrix}

and the statistical properties of these expansions are given by the ergodic properties of T. The study of these ergodic properties has been hindered by the fact that there is no known formula for the absolutely continuous T -invariant probability measure μ. The purpose of this note is to provide very accurate approximate ergodic properties of the Bolyai-Rényi expansion with respect to μ.

Firstly, we compute the corresponding metric entropy h(μ), which is also equal to the Lyapunov exponent of μ. The entropy reflects the average “quality” of the expansion (see Proposition 1 and Corollary 2). We derive (Theorem 1) the estimate h(μ) = 1.05631307402, and prove that this is within 1.3 × 10^{−7} of the true entropy. We also give an empirical estimate h(μ) = 1.056313074. By an empirical estimate we mean that we are confident this value is correct up to the accuracy given, and that this is strongly supported by numerical evidence (see below for more details of our criterion).

Secondly, we compute the frequencies of the digits 0, 1 and 2 occurring in almost every expansion. In Theorem 2 we prove that

freq(0) = 0.46407962943 ± 2.9 × 10^{−8}
freq(1) = 0.3044190449 ± 1.6 × 10^{−8}
freq(2) = 0.2315013256 ± 1.1 × 10^{−8}

where freq(i) = lim_{N→+∞} \frac{1}{N} #\{1 ≤ k ≤ N : a_{k} = i\}. From this we derive (Corollary 3) the asymptotic arithmetic mean of the digits.

Thirdly, it is well known that the correlation function c_{n}(f, g) = ∫ f · T^{n}g dμ − ∫ f dμ ∫ g dμ of two analytic functions f, g converges to zero at some exponential rate O(α^{n}), 0 < α < 1, which can be chosen independent of f, g. We estimate (Theorem 3) this rate of decay of correlations to be α = 0.40582, and prove rigorously that this is within 0.000076 of the true value.

In our calculations we exploit the analyticity of T, and combine this with some ideas from thermodynamic formalism. To compute the entropy we use the Rohlin-Pesin formula h(μ) = ∫ log |T′|dμ. By computing the periodic points of T we derive an expression for h(μ) as the ratio of two rapidly converging series expansions. The approximations to h(μ) are given by working with the corresponding order-n truncated series. We obtain our empirical estimate by choosing n as large as is computationally feasible, then computing both the order-n and the order-(n − 1) truncations. We then compare these estimates, and retain those digits which agree. The (non-rigorous) justification for this is an empirical observation of the rapid convergence rate (see Table 1). To obtain our rigorous estimate, we combine an exact calculation of the first 12 terms in each series, with a rigorous estimate on the tail of the series from the 13^{th} term onwards. To obtain this tail estimate we give explicit bounds on the rate of decay of terms in the series expansions. The computation of digit frequencies is similar, though here we integrate characteristic functions rather than log |T′|.

It is well known that the rate of correlation decay is by the second largest eigenvalue of the Perron-Frobenius operator L (defined in section 3). When L acts on a suitable analytic function space, its eigenvalues are given by the reciprocals of the zeros z of its Fredholm determinant. This determinant is an entire function, so by computing the zeros of its polynomial truncations we obtain rapid approximations to the true rate of correlation decay.

The article is organised as follows. In Section 2 we recall some familiar properties of f-expansions, and discuss a few examples. In Section 3 we specialise to the Bolyai-Rényi map, considering its extension to certain discs in the complex plane, and introducing a family of transfer operators. In Section 4 we show that these operators are nuclear, and obtain explicit bounds on the coefficients in the power series expansions of their Fredholm determinants. Sections 5, 6 and 7 are devoted, respectively, to the proofs of the results on the entropy, digit frequencies, and rate of correlation decay.

2. f-expansions

To each of the f-expansions described in the introduction, we can associate an expanding map T of the unit interval I, defined by T (x) = {f^{−1}(x)}, where {·} denotes the fractional part.

Lemma 1. Such expanding maps always possess an invariant probability measure μ, absolutely continuous with respect to Lebesgue measure λ [LaY]. This absolutely continuous invariant measure is unique.

Let h(μ) denote the entropy of μ. Let a_{k} = a_{k}(x) denote the k^{th} digit in the f-expansion of x, and denote I_{i} = {x : a_{1}(x) = i}.

Proposition 1.

(1) freq(i) := lim_{N→+∞} \frac{1}{N} #\{1 ≤ k ≤ N : a_{k}(x) = i\} = μ(I_{i}) for almost all points x,

(2) If we denote x_{N} = f (a_{1} + f(a_{2} + … + f(a_{N}) …)), N ≥ 1, then for any ε > 0, and almost all points x, we have that |x − x_{N}| = O(e^{−N(h(μ)−ε)}).

Proof. Part (1) is a consequence of the Birkhoff Ergodic Theorem, while part (2) follows from the Shannon-McMillan-Breiman theorem, (see for example [Wal]).

Example 1. In the case of continued fractions we have the Gauss map T (x) = {1/x}, a Markov map with infinitely many branches. The invariant density dμ/dλ(x) = (log 2(1 + x))^{−1} is known explicitly, and this allows an exact calculation of the corresponding metric entropy as \frac{π^{2}}{6 log 2}, see [Sch].

Example 2. The β-transformation T (x) = {βx} is in general not a Markov map (see [Bl]), though the function dμ/dλ(x) = ∑_{n=0}^{∞} β^{−n}χ_{{z:z<T^{n}(1)}}(x) gives an exact formula for the (non-normalised) invariant density of μ (see [Pa]). The entropy is simply log β, by the Rohlin-Pesin formula [Ro], [Pe].

The symbolic dynamics of the Bolyai-Rényi map T (x) = {(x + 1)^{2} − 1} is particularly simple, a full shift on a three letter alphabet. However, despite the existence of an absolutely continuous invariant measure μ, an exact formula for its density function is not known. Consequently, other statistical information such as the frequency of digits, the metric entropy, the rate of correlation decay, etc. are also unknown.

In the Bolyai-Rényi case, T is an expanding Markov map with respect to the partition consisting of intervals I_{0} = [0, \sqrt{2}−1], I_{1} = [\sqrt{2}−1, \sqrt{3}−1], I_{2} = [\sqrt{3}−1, 1]. The corresponding inverse branches T_{j} : I → I_{j} are given by

T_{j}(x) = \sqrt{j + 1 + x} − 1.

Given x ∈ [0, 1], the sequence of digits a_{n} = a_{n}(x) is generated by setting r_{0} = x, then defining recursively

r_{n+1} = {(1 + r_{n})^{2} − 1} a_{n+1} = [(1 + r_{n})^{2} − 1].

Equivalently, for x ∈ [0, 1], we have r_{n} = T^{n}x, and a_{n} is such that T^{n−1}x ∈ I_{a_{n}}.

The sequence of digits is unique unless some iterate of x lies on the endpoint of a piece of the Markov partition. The set of such points is countable, and in particular of zero Lebesgue measure, so we freely ignore them in the sequel.

3. Preliminaries

Our analysis will involve the periodic points of the Bolyai-Rényi map T. The two endpoints 0 and 1 are fixed by T, as is the golden ratio

γ = −1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + …}}}

For a given string \underline{j} = j_{1} … j_{n} ∈ \{0, 1, 2\}^{n} of length |\underline{j}| = n, we define the contraction T\underline{j} to be the composition T_{j_{1}} · . . . · T_{j_{n}}. The algebraic number x\underline{j}, defined as the fixed point of T\underline{j}, is a period-n point under T, and all period-n points arise in this way.

For a general closed disc D ⊂ ℂ, let A_{∞}(D) denote the Banach space of complexvalued functions which are continuous on D and holomorphic on int(D), equipped with the supremum norm.

Now let us fix the disc D to be centred at 1/2, of radius \sqrt{3}/2 ≈ 0.866. Each branch T_{j} has an analytic extension to D, which we also denote by T_{j}. Its derivative extends to

T′_{j}(z) = \frac{1}{2\sqrt{z + j + 1}} = \frac{1}{2(T_{j}z + 1)}.

Define the composition-type operators L_{j}w(z) = w(T_{j}z)T′_{j}(z), and then the Perron-Frobenius operator L : A_{∞}(D) → A_{∞}(D) by

Lw(z) = ∑_{j=0}^{2} L_{j}w(z) = ∑_{j=0}^{2} \frac{w(T_{j}z)}{2\sqrt{z + j + 1}} = ∑_{j=0}^{2} \frac{w(T_{j}z)}{2(T_{j}z + 1)}.  (3.1)

We will be interested in integrating various analytic functions g with respect to the measure μ. For example, the choice g(x) = log |T′(x)| = log(2x+2) (see Section 5) gives the entropy h(μ).

Given any function g ∈ A_{∞}(D), for all t ∈ ℂ we also have e^{tg·T_{j}} ∈ A_{∞}(D), for each j = 0, 1, 2, so define transfer operators L_{g,t} on A_{∞}(D) by

L_{g,t}w(z) = ∑_{j=0}^{2} \frac{e^{tg(T_{j}z)}w(T_{j}z)}{2\sqrt{z + j + 1}} = ∑_{j=0}^{2} \frac{e^{tg(T_{j}z)}w(T_{j}z)}{2(T_{j}z + 1)}.  (3.2)

The closed disc D′, centred at 1/2 and radius \frac{3}{2}(\sqrt{3} − 1) ≈ 1.098, contains the smaller disc D. The disc D′ is also contained in each of the (local) inverse images T_{j}^{−1}D. In other words,

T_{1}(D′) ∪ T_{2}(D′) ∪ T_{3}(D′) ⊂ D.

Moreover, each of the derivatives T′_{j} extend analytically to the larger disc D′.

Figure. The disks D and D′, and the images T_{i}(D′), i = 1, 2, 3.

The crucial consequence of this is that if w ∈ A_{∞}(D) then L_{g,t}w ∈ A_{∞}(D′). That is,

L_{g,t}A_{∞}(D) ⊂ A_{∞}(D′).

So the operator L_{g,t} improves the regularity of functions it operates on: if w is merely analytic on D, then its image L_{g,t}w is analytic on the larger disc D′.

We will exploit this in the proof of Proposition 3, where a circle of integration will be pushed arbitrarily close to the boundary of D′. A key quantity will be the ratio

θ = \frac{1}{3 − \sqrt{3}} ≈ 0.788675 …

of the radii of the concentric discs D ⊂ D′. We call θ the contraction ratio. For ε > 0 it will be convenient to denote

θ_{ε} = \frac{\sqrt{3}/2}{\frac{3}{2}(\sqrt{3} − 1) − ε},

and note that θ_{ε} ↘ θ as ε → 0.

Remark. Our disc D was chosen so as to minimise the contraction ratio θ.

Although the density function ϱ := dμ/dλ for the absolutely continuous invariant measure μ is not explicitly known, we have the following result.

Proposition 2. The density function ϱ = dμ/dλ is real analytic on the interval [0, 1]. Moreover, it has a holomorophic extension to the cut plane ℂ \ (−∞, −1].

Proof. The fact that T is piecewise expanding, piecewise onto and piecewise analytic (all with respect to our Markov partition) means that ϱ is itself real analytic. This follows immediately from the well-known fact that ϱ is the fixed point of the Perron-Frobenius operator L, and that this linear operator preserves the space of real analytic functions defined on I (see [PJ] for more details).

Let us suppose that the function ϱ is analytic on a domain D ⊂ ℂ containing [0, 1]. The three maps T_{j} are analytic on the cut plane ℂ \ (−∞, −1]. Moreover, the same is true of the weight functions 1/(2\sqrt{z + j + 1}) in definition (3.1).

If N_{ε}((−∞, −1]) denotes the ε-ball around the half-line (−∞, −1] then define B(R, ε) = {z ∈ ℂ : |z| < R} \ N_{ε}((−∞, −1]). For n sufficiently large, each T_{j}^{n}B(R, ε) lies inside the domain D, thus the compositions ϱ ∘ T_{j}^{n} are holomorphic on B(R, ε), for j = 0, 1, 2. Therefore ϱ = L^{n}ϱ is also holomorphic on B(R, ε). Letting R → ∞ and ε → 0 gives the result.

4. Fredholm determinants and nuclear operators

Letting g^{n}(x) = g(x)+g(T x)+… g(T^{n−1}x), we define the following two-variable function F = F_{g} in terms of our function g and the periodic points of T,

F (z, t) = F_{g}(z, t) = exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{|\underline{i}|=n} \frac{T ′ ; \underline{i}(x\underline{i}) exp (tg^{n}(x\underline{i}))}{1 − T ′ ; \underline{i}(x\underline{i})}) .  (4.1)

If D_{1}F, D_{2}F denote first and second partial derivatives, respectively, then we have

Lemma 2.

∫ g dμ = \frac{D_{2}F (1, 0)}{D_{1}F (1, 0)}

The proof of this Lemma is contained in [PJ]. The idea is to relate the integral to a certain derivative of topological pressure, which in turn is related to a zero of F.

Proposition 3. For each t ∈ ℂ there exist sequences of functions v_{n} ∈ A_{∞}(D) and functionals l_{n} ∈ A_{∞}(D)^{∗} such that, for all w ∈ A_{∞}(D),

L_{g,t}w = ∑_{n=1}^{∞} l_{n}(w)v_{n}

where ||l_{n}||_{∞}||v_{n}||_{∞} = O(θ^{n}), and θ = \frac{1}{3−\sqrt{3}} ≈ 0.788675 . . ..

Proposition 3, together with the definition of a nuclear operator (see [Gr], [PJ]), immediately gives us:

Corollary 1. For each t ∈ ℂ, the operator L_{g,t} is nuclear.

Proof of Proposition 3. Let Γ_{ε} be the circle centred at 1/2, of radius \frac{3}{2}(\sqrt{3} − 1) − ε, where ε > 0. For ε sufficiently small we note that Γ_{ε} is outside of D but is just inside D′.

For w ∈ A_{∞}(D) we know that L_{g,t}w ∈ A_{∞}(D′) (i.e. is analytic on the larger disc D′). So using Cauchy’s theorem we have

L_{g,t}w(z) = \frac{1}{2πi} ∫_{Γ_{ε}} \frac{(L_{g,t}w)(ξ)}{ξ − z}dξ

= \frac{1}{2πi} ∫_{Γ_{ε}} \frac{(L_{g,t}w)(ξ)}{ξ − 1/2} (1 − \frac{z − 1/2}{ξ − 1/2})^{−1} dξ

= ∑_{n=0}^{∞} l_{n}(w)v_{n}(z),

where

v_{n}(z) = (z − 1/2)^{n}  and  l_{n}(w) = \frac{1}{2πi} ∫_{Γ_{ε}} \frac{(L_{g,t}w)(ξ)}{(ξ − 1/2)^{n+1}} dξ.

Given that z ∈ D and ξ ∈ Γ, we have

|(z − 1/2)^{n}| ≤ (\frac{\sqrt{3}}{2})^{n} and |ξ − 1/2|^{n+1} = (\frac{3}{2}(\sqrt{3} − 1) − ε)^{n+1} .

Therefore, for all sufficiently small ε > 0, we have the estimate

\begin{aligned}||l_{n}||_{∞}||v_{n}||_{∞} & ≤ \frac{||L_{g,t}||_{∞}}{2π(\frac{3}{2}(\sqrt{3} − 1) − ε)} ( \frac{\sqrt{3}/2}{\frac{3}{2}(\sqrt{3} − 1) − ε})^{n} \\ & = K_{1}||L_{g,t}||_{∞}θ_{ε}^{n}, \end{aligned}  (4.2)

where K_{1} := 0.1449397969 . . ..

Thus ||l_{n}||_{∞}||v_{n}||_{∞} = O(θ^{n}), as claimed.

Now we show that the integral ∫ g dμ can be effectively approximated.

Proposition 4. For each g ∈ A_{∞}(D), there exists a real-valued sequence S_{n}(g), whose n^{th} term is computable from the periodic points up to period n, such that

∫ g dμ = S_{n}(g) + O(θ^{n(n+1)/2}),

where θ = \frac{1}{3−\sqrt{3}} ≈ 0.788675 . . ..

Proof. Here we will only sketch the idea of the proof. Full details can be found in [PJ], and explicit estimates are carried out in sections 5 and 6, for particular choices of functions g.

The terms

∑_{|\underline{i}|=n} \frac{T ′ ; \underline{i}(x\underline{i}) exp (tg^{n}(x\underline{i}))}{1 − T ′ ; \underline{i}(x\underline{i})}

in (4.1) are precisely the traces of the n^{th} iterate of our nuclear operator L_{g,t}, which means that F (z, t) is in fact the Fredholm determinant of L_{g,t} (see [Gr]), and in particular is an entire function of both z ∈ ℂ and t ∈ ℂ.

Consequently there is an alternative form for F (z, t) as the power series

F (z, t) = 1 + ∑_{n=1}^{∞} C_{n}(t)z^{n},

whose coefficients are given (see [Gr], p. 17) by

C_{n}(t) = ∑_{i_{1}<...<i_{n}} λ_{i_{1}} … λ_{i_{n}}Det [(m_{i_{α}}(u_{i_{β}}))_{1≤α,β≤n}] ,

where m_{i} := l_{i}/||l_{i}||_{∞}, u_{i} := v_{i}/||v_{i}||_{∞}, and λ_{i} := ||l_{i}||_{∞}||v_{i}||_{∞} (all supremum norms are with respect to the disc D).

Since the entries in the matrix (m_{i_{α}}(u_{i_{β}}))_{1≤α,β≤n} are less than one in absolute value, its determinant is dominated in modulus by n^{n/2}, by a theorem of Hadamard [Had]. Our estimate (4.2) on the rate of decay of the λ_{i} implies

\begin{aligned}∑_{i_{1}<...<i_{n}} λ_{i_{1}} … λ_{i_{n}} & ≤ (K_{1}||L_{g,t}||_{∞})^{n} ∑_{i_{1}<...<i_{n}} θ^{i_{1}+...+i_{n}} \\ & = (K_{1}||L_{g,t}||_{∞})^{n} \frac{θ^{n(n+1)/2}}{(1 − θ) … (1 − θ^{n})}. \end{aligned}

Combining these gives the estimate

|C_{n}(t)| ≤ \frac{n^{n/2}(K_{1}||L_{g,t}||_{∞})^{n}θ^{n(n+1)/2}}{(1 − θ) … (1 − θ^{n})} .  (4.3)

Since ∏_{n=1}^{∞}(1 − θ^{n}) > 0, then the expression in (4.3) is O(θ^{n(n+1)/2}), (for more details see Lemma 8, [PJ]).

From this we deduce the same decay rate O(θ^{n(n+1)/2}) for the power series coefficients of the partial derivatives D_{1}F and D_{2}F (see sections 5 and 6 for explicit estimates of these coefficients).

By Lemma 1 we can define S_{n}(g) as the ratio of the degree-n polynomial truncations of D_{1}F and D_{2}F (evaluated at z = 1, t = 0).

Finally, we can use the series expansion of the exponential function in (4.1) to obtain an alternative form of the power series expansion for F, where this time the coefficients C_{n}(t) are in terms of the periodic points of T. Namely we have

C_{n}(t) = ∑_{\begin{matrix}(j_{1},... ,j_{m}) \\ j_{1}+...+j_{m}=n \end{matrix}} \frac{(−1)^{m}}{m!} a_{j_{1}}(t) … a_{j_{m}}(t),  (4.4)

where

a_{j}(t) = ∑_{|\underline{i}|=j} \frac{T ′ ; \underline{i}(x\underline{i}) exp (tg^{n}(x\underline{i}))}{j(1 − T ′ ; \underline{i}(x\underline{i}))}.  (4.5)

Thus the S_{n}(g) are also computable from the periodic points up to period n, as claimed.

5. Entropy

In this section we compute the entropy of the absolutely continuous invariant measure μ. To do this we fix our choice of analytic function g to be g(x) = log(2 + 2x), since this is the analytic extension of log |T′(x)|. By the Rohlin-Pesin identity (see [Ro], [Pe]) we know that h(μ) = ∫ log |T′| dμ.

Theorem 1. The entropy h(μ) of the absolutely continuous invariant measure μ for the Bolyai-Rényi expansion is h(μ) = 1.05631307402 ± 1.3 × 10^{−7}.

Moreover, we have the empirical estimate h(μ) = 1.056313074 = log 2.875748746.

Proof.

Choosing g(x) = log |T′(x)| = log(2 + 2x), we define F (z, t) as in (4.1). We have series expansions

D_{1}F (1, 0) = ∑_{n=1}^{∞} nC_{n}(0) , D_{2}F (1, 0) = ∑_{n=1}^{∞} C′_{n}(0).

Using the formulae (4.4), (4.5) we can exactly compute the first 12 terms (our only limitation here is available computer power) in both of the above series. This gives us the order-12 approximations:

\begin{matrix}c_{1} := ∑_{n=1}^{12} nC_{n}(0) = −0.393317995898306598279 …, and \\ c_{2} := ∑_{n=1}^{12} C′_{n}(0) = −0.4154669413176785127047 …. \end{matrix}

By Lemma 2, this gives us the order-12 approximation

h(μ) = \frac{D_{2}F (1, 0)}{D_{1}F (1, 0)} ≈ \frac{c_{2}}{c_{1}} = 1.05631307402750669584133 …

To estimate the accuracy of this approximation we introduce the error terms ε_{i} := |D_{i}F (1, 0) − c_{i}|, for i = 1, 2.

First we estimate ε_{1}. To do this we will bound the norm ||L_{g,t}||_{∞} appearing in (4.2), for the case t = 0. That is, we will bound the norm ||L||_{∞} of the Perron-Frobenius operator. We have

||L_{g,0}||_{∞} = ||L||_{∞} ≤ ∑_{j=0}^{2} sup_{z∈D} \frac{1}{2\sqrt{z + j + 1}}.

By the maximum modulus theorem, each of the above suprema is attained by some z ∈ ∂D, the circle of radius \sqrt{3}/2 around 1/2. In fact it is easily seen that in each case the supremum is attained (uniquely) at the point z = (1 − \sqrt{3})/2, so that

||L_{g,0}||_{∞} = ||L||_{∞} ≤ \frac{1}{2\sqrt{\frac{1}{2}(3 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(5 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(7 − \sqrt{3})}}
≈ 0.6279630302 + 0.3911536204 + 0.308080502
= 1.327197153.

Therefore we have

K_{2} := K_{1}||L||_{∞} ≤ 0.1449397969 × 1.327197153 = 0.1923636858,

so that (4.3) gives

\begin{matrix}ε_{1} = & | ∑_{n=13}^{∞} nC_{n}(0)| ≤ ∑_{n=13}^{∞} \frac{n^{n/2}nK_{2}^{n}θ^{n(n+1)/2}}{(1 − θ) … (1 − θ^{n})} \\ = & 0.0000000080413605760125 … ≈ 8.04136 × 10^{−9} \end{matrix}

Now we make the more delicate estimate of the error ε_{2}. This will involve estimating ||L_{g,t}||_{∞}, for general t ∈ ℂ. Since our function is g(x) = log(2 + 2x), using (3.2) we have, for any t ∈ ℂ,

||L_{g,t}||_{∞} ≤ ∑_{j=0}^{2} sup_{z∈D} |\frac{(1 + 2\sqrt{j + 1 + z})^{−t}}{2\sqrt{j + 1 + z}} |

= ∑_{j=0}^{2} sup_{z∈∂D} |\frac{(1 + 2\sqrt{j + 1 + z})^{−t}}{2\sqrt{j + 1 + z}} |

= ∑_{j=0}^{2} max_{0≤α≤1} |\frac{(1 + 2\sqrt{j + \frac{3}{2} + \frac{\sqrt{3}}{2}e^{2πiα}})^{−t}}{2\sqrt{j + \frac{3}{2} + \frac{\sqrt{3}}{2}e^{2πiα}}} |,

(5.1)

again by the maximum modulus principle.

Now we define

J(δ) = K_{1} max_{|t|=δ} ∑_{j=0}^{2} max_{0≤α≤1} |\frac{(1 + 2\sqrt{j + \frac{3}{2} + \frac{\sqrt{3}}{2}e^{2πiα}})^{−t}}{2\sqrt{j + \frac{3}{2} + \frac{\sqrt{3}}{2}e^{2πiα}}} |.  (5.2)

By (5.1) we know J(δ) ≥ K_{1}||L_{g,t}||_{∞} for all t ∈ ℂ with |t| = δ.

Since F_{g}(z, t) is entire in t, by Cauchy’s theorem we can estimate C′_{n}(0) by integration around some suitable circle centred at 0. If this circle is of radius δ > 0, then (4.3), (5.1), (5.2) give

|C′_{n}(0)| = |\frac{1}{2πi} ∫_{|t|=δ} \frac{C_{n}(t)}{t^{2}}dt|

≤ \frac{max_{|t|=δ} |C_{n}(t)|}{δ^{2}2π}

≤ \frac{n^{n/2}J(δ)^{n}θ^{n(n+1)/2}}{δ^{2}2π(1 − θ) … (1 − θ^{n})}.

We then have

ε_{2} = | ∑_{n=13}^{∞} C′_{n}(0)| ≤ ∑_{n=13}^{∞} \frac{n^{n/2}J(δ)^{n}θ^{n(n+1)/2}}{δ^{2}2π(1 − θ) … (1 − θ^{n})},  (5.3)

for all δ > 0, and it remains to make a suitable choice of δ to obtain the optimal upper bound on ε_{2}.

Choosing δ = 0.131 (this gives close to the optimal estimate, given that we are using order-12 truncations) we can compute numerically that

\begin{aligned}J(0.131) & = 0.2240247382219096 = K_{1} × 1.545639934375747 \\ & = K_{1} (0.7114266780771788 + 0.4618830103302341 + 0.3723302459683343) . \end{aligned}

(We remark that the suprema in the definition of J(·) are achieved for the value t = −δ = −0.131, and the value α = 1/2 (corresponding to the point z = 1/2 − \sqrt{3}/2 ∈ ∂D) for each j = 0, 1, 2).

In particular, for this choice of δ, (5.3) yields the estimate

ε_{2} ≤ 0.00000004174688467881798 ≈ 4.1746 × 10^{−8}.

We then have

|\frac{c_{2}}{c_{1}} − \frac{D_{2}F (1, 0)}{D_{1}F (1, 0)}| ≤ \frac{|c_{2}|ε_{1} + |c_{1}|ε_{2}}{|c_{1}|(|c_{1}| − ε_{1})} ≤ 0.0000001277365402532839 … ≈ 1.277×10^{−7},

so our order-12 approximation is indeed within 1.3 × 10^{−7} of the true entropy.

To establish the empirical estimate, we compute the order-11 approximation (see Table 1) to the entropy as 1.056313074656433934604373.

Comparing this with the order-12 approximation, and observing (see Table 1) the speed of convergence, we obtain the empirical estimate h(μ) = 1.056313074.

n

5  1.05816770765660556718251676
6  1.0561527030666136783615544
7  1.0563251436448792708699773
8  1.0563120810658188337599628
9  1.0563131596010201356828930
10  1.056313066925865599688266
11  1.056313074656433934604373
12  1.05631307402750669584133

Table 1. Successive approximations to the entropy h(μ) using periodic points up to period n

Remark. In [BDK] it was conjectured (Conjecture 6.1) that h(μ) ≈ 1.0545. These authors use a statistical approach based on the use of the Shannon-McMillan-Breiman theorem, which is very different in nature from our approach.

Corollary 2. If x_{N} = f (a_{1} + f(a_{2} + … + f(a_{N}) …)), with f(x) = \sqrt{1 + x} − 1, then for Lebesgue almost all points x we have |x − x_{N}| = O(0.34773557^{N}).

Proof. By Theorem 1, the true entropy h(μ) is at least 1.05631294, so by choosing some suitably small ε > 0 in part (2) of Proposition 1, we see that the exponential contraction rate of |x − x_{N}| is O(0.34773557^{N}), for almost every x.

6. Frequency of digits

By part (1) of Proposition 1, the frequencies with which the digits 0, 1, 2 appear in the Bolyai-Rényi expansion are well-defined (for Lebesgue almost every point), and given by integrating the characteristic functions χ_{I_{i}}, i = 0, 1, 2, with respect to the invariant measure μ. In this section we will accurately compute these frequencies.

Our method involves a slight modification of the one used to compute the entropy. Note that each χ_{I_{i}} does not belong to A_{∞}(D). However, its restriction to each of the intervals I_{j} clearly is analytic, for j = 0, 1, 2. More formally, χ_{I_{i}} belongs to the space of functions which are analytic on the disjoint union ∐_{j=0}^{2} I_{j}. Indeed, taking the three separate analytic extensions of χ_{I_{i}}|_{I_{j}}, (j = 0, 1, 2), to D gives a function which is analytic on the disjoint union ∐_{j=0}^{2} D_{j}, where each D_{j} is a copy of D. This new function is simply χ_{D_{i}}. Moreover, the Fredholm determinant (defined by (4.1)) for g = χ_{I_{i}} coincides with that for g = χ_{D_{i}}, since all period-n points x\underline{i} lie on I. Thus for calculational purposes we can work with the function χ_{D_{i}} rather than χ_{I_{i}}.

The important advantage of this is the following. Since each χ_{D_{i}} ∘ T_{j} = 0 or 1, then L_{t}^{(i)} := L_{χ_{D_{i}} ,t} acts on A_{∞}(D), and its image is contained in the space A_{∞}(D′) of functions analytic on the larger disc D′ (by the same arguments as at the end of section 3). Consequently the analysis in section 4, and in particular the estimate (4.3), remains valid for g = χ_{D_{i}}.

Explicitly, our transfer operators L_{t}^{(i)} := L_{χ_{D_{i}} ,t} are defined by

L_{t}^{(i)}w(z) = \frac{e^{t}w(T_{i}z)}{2\sqrt{z + i + 1}} + ∑_{{j=0 \atop j≠i}}^{2} \frac{w(T_{j}z)}{2\sqrt{z + j + 1}}.  (6.1)

Theorem 2. For Lebesgue almost every x ∈ I, the frequency of occurrence of the digits 0, 1, 2 in the Bolyai-Rényi expansion of x are

freq(0) = 0.46407962943 ± 2.9 × 10^{−8}

freq(1) = 0.3044190449 ± 1.6 × 10^{−8}
freq(2) = 0.2315013256 ± 1.1 × 10^{−8}

Moreover, we have the empirical estimates freq(0) = 0.46407962943

freq(1) = 0.3044190449
freq(2) = 0.2315013256

Proof. By analogy with (4.1) we introduce the entire functions

F^{(j)}(z, t) = exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{|\underline{i}|=n} \frac{T ′ ; \underline{i}(x\underline{i}) exp (tχ_{I_{j}}^{n}(x\underline{i}))}{1 − T ′ ; \underline{i}(x\underline{i})})

= exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{|\underline{i}|=n} \frac{T ′ ; \underline{i}(x\underline{i}) exp (tχ_{D_{j}}^{n}(x\underline{i}))}{1 − T ′ ; \underline{i}(x\underline{i})})

= 1 + ∑_{n=1}^{∞} C_{n}^{(j)}(t)z^{n},

(6.2)

���

ERGODIC PROPERTIES OF THE BOLYAI-RÉNYI EXPANSION 13

for j = 0, 1, 2.

By Proposition 1 and Lemma 2 we can identify

freq(j) = ∫ χ_{I_{j}}dμ = \frac{D_{2}F^{(j)}(1, 0)}{D_{1}F^{(j)}(1, 0)},

where we have the series expansions

D_{1}F^{(j)}(1, 0) = ∑_{n=1}^{∞} nC_{n}^{(j)}(0) , D_{2}F^{(j)}(1, 0) = ∑_{n=1}^{∞} C_{n}^{(j)}′(0).

Note in fact that D_{1}F^{(j)}(1, 0) is independent of j, so as in the proof of Theorem 1 we have the order-12 approximation:

c_{1} := ∑_{n=1}^{12} nC_{n}^{(j)}(0) = −0.393317995898306598279,

for j = 0, 1, 2.

Defining the error term ε_{1} := |D_{1}F^{(j)}(1, 0) − c_{1}| (which is independent of j), by the same argument as in Theorem 1 we obtain the upper bound

ε_{1} ≤ 0.0000000080413605760125 … ≈ 8.04136 × 10^{−9}

The relevant order-12 approximations for D_{2}F^{(j)}(1, 0) are respectively

c_{2}^{(0)} := ∑_{n=1}^{12} C_{n}^{(0)}′(0) = −0.182530869785549059806378048463367126373 …,

c_{2}^{(1)} := ∑_{n=1}^{12} C_{n}^{(1)}′(0) = −0.11973348866379947317802423837617903111 …,

c_{2}^{(2)} := ∑_{n=1}^{12} C_{n}^{(2)}′(0) = −0.0910536374334888938007591366192137185 …

By Lemma 2, this gives us the order-12 approximations

freq(0) = \frac{D_{2}F^{(0)}(1, 0)}{D_{1}F^{(0)}(1, 0)} ≈ \frac{c_{2}^{(0)}}{c_{1}} = 0.4640796294323204473785420774671837071 …

freq(1) = \frac{D_{2}F^{(1)}(1, 0)}{D_{1}F^{(1)}(1, 0)} ≈ \frac{c_{2}^{(1)}}{c_{1}} = 0.3044190449265811915843610512988279987 …

freq(2) = \frac{D_{2}F^{(2)}(1, 0)}{D_{1}F^{(2)}(1, 0)} ≈ \frac{c_{2}^{(2)}}{c_{1}} = 0.2315013256017684253229890778051297912 …

To estimate the accuracy of these approximation we introduce the error terms ε_{2}^{(j)} := |D_{2}F^{(j)}(1, 0) − c_{2}^{(j)}|, for j = 0, 1, 2.

To estimate the ε_{2}^{(j)}, we need to estimate ||L_{t}^{(j)}||_{∞}. By analogy with the techniques used in the proof of Theorem 1, we define

J_{(0)}(δ) = K_{1} max_{|t|=δ} (max_{z∈∂D} |\frac{e^{t}}{2\sqrt{z + 1}}| + max_{z∈∂D} |\frac{1}{2\sqrt{z + 2}}| + max_{z∈∂D} |\frac{1}{2\sqrt{z + 3}}|)

= K_{1} (\frac{e^{δ}}{2\sqrt{\frac{1}{2}(3 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(5 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(7 − \sqrt{3})}})

(6.3)

Note that J_{(0)}(δ) ≥ K_{1}||L_{t}^{(0)}||_{∞} for all t ∈ ℂ with |t| = δ.

Defining J_{(1)}(δ), J_{(2)}(δ) analogously, we derive the explicit expressions

J_{(1)}(δ) = K_{1} (\frac{1}{2\sqrt{\frac{1}{2}(3 − \sqrt{3})}} + \frac{e^{δ}}{2\sqrt{\frac{1}{2}(5 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(7 − \sqrt{3})}}) ,

J_{(2)}(δ) = K_{1} (\frac{1}{2\sqrt{\frac{1}{2}(3 − \sqrt{3})}} + \frac{1}{2\sqrt{\frac{1}{2}(5 − \sqrt{3})}} + \frac{e^{δ}}{2\sqrt{\frac{1}{2}(7 − \sqrt{3})}}) .

As in the proof of Theorem 1, we can estimate C_{n}^{(j)}′(0) by integrating around a circle of radius δ, centred at 0. By (4.3), (6.3) we have

|C_{n}^{(j)}′(0)| = |\frac{1}{2πi} ∫_{|t|=δ} \frac{C_{n}^{(j)}(t)}{t^{2}}dt|

≤ \frac{max_{|t|=δ} |C_{n}^{(j)}(t)|}{δ^{2}2π}

≤ \frac{n^{n/2}J_{(j)}(δ)^{n}θ^{n(n+1)/2}}{δ^{2}2π(1 − θ) … (1 − θ^{n})}.

We then have

ε_{2}^{(j)} = | ∑_{n=13}^{∞} C_{n}^{(j)}′(0)| ≤ ∑_{n=13}^{∞} \frac{n^{n/2}J_{(j)}(δ)^{n}θ^{n(n+1)/2}}{δ^{2}2π(1 − θ) … (1 − θ^{n})},  (6.4)

for all δ > 0, and it remains to make a suitable choice of δ to obtain the optimal upper bound on ε_{2}^{(j)}. The choise of δ will be different for each j = 0, 1, 2.

For j = 0 we choose δ = 0.280538, which gives

J_{(0)}(0.280538) = 0.22183874550802618353,

and (6.4) gives the upper bound

ε_{2}^{(0)} ≤ 0.0000000080094010700419 … ≈ 8.0094 × 10^{−9}.

This gives us

|\frac{c_{2}^{(0)}}{c_{1}} −\frac{D_{2}F (1, 0)}{D_{1}F (1, 0)}| ≤ \frac{|c_{2}^{(0)}|ε_{1} + |c_{1}|ε_{2}^{(0)}}{|c_{1}|(|c_{1}| − ε_{1})} ≤ 0.000000029851756260 … ≈ 2.9851×10^{−8},

so our order-12 approximation is indeed within 2.9 × 10^{−8} of the true frequency freq(0).

For j = 1 we choose δ = 0.4, which gives J_{(1)}(0.4) = 0.22024706058611192697, and (6.4) gives the upper bound

ε_{2}^{(1)} ≤ 0.000000003586306645535231 … ≈ 3.5863 × 10^{−9}.

We then have

|\frac{c_{2}^{(1)}}{c_{1}} −\frac{D_{2}F (1, 0)}{D_{1}F (1, 0)}| ≤ \frac{|c_{2}^{(1)}|ε_{1} + |c_{1}|ε_{2}^{(1)}}{|c_{1}|(|c_{1}| − ε_{1})} ≤ 0.0000000153419119854 … ≈ 1.5341×10^{−8},

so our order-12 approximation is indeed within 1.6 × 10^{−8} of the true frequency freq(1).

For j = 2 we choose δ = 0.47, which gives J_{(2)}(0.47) = 0.21915530173992111521, and (6.4) gives the upper bound

ε_{2}^{(2)} ≤ 0.000000002434474277421805 … ≈ 2.4344 × 10^{−9}.

We then have

|\frac{c_{2}^{(2)}}{c_{1}} −\frac{D_{2}F (1, 0)}{D_{1}F (1, 0)}| ≤ \frac{|c_{2}^{(2)}|ε_{1} + |c_{1}|ε_{2}^{(2)}}{|c_{1}|(|c_{1}| − ε_{1})} ≤ 0.0000000109226123468 … ≈ 1.0922×10^{−8},

so our order-12 approximation is indeed within 1.1 × 10^{−8} of the true frequency freq(2).

The empirical estimate for each freq(j) is given by comparing the 11^{th} and 12^{th} order approximations in Tables 2, 3 and 4, and retaining those digits which coincide.

n
5  0.46435909501454506686307566929087458078
6  0.46406049499842454303522745347088830886
7  0.46408186730089402715550697909797236095
8  0.46407938130821526467334701452249103314
9  0.46407965347225846507480290244178162010
10  0.4640796272949911418861341265524892711
11  0.4640796296310131100783392003010399183
12  0.4640796294323204473785420774671837071

Table 2. Successive approximations to freq(0) using periodic points up to period n

Corollary 3. For almost every x ∈ I, with digit sequence a_{1}, a_{2}, …, the asymptotic arithmetic mean is

lim_{n→∞} \frac{1}{n} (a_{1} + … + a_{n}) = 0.767421696 ± 0.56 × 10^{−8}

n

5  0.30503305810398604203711128821431036127
6  0.30436504181880606086969695969237938890
7  0.30442298977744286102431668914647584683
8  0.30441873023402450744848124413377540095
9  0.30441907163931442404009449037405814340
10  0.3044190427311780262387140577206270104
11  0.3044190451195828345302730034394649730
12  0.3044190449265811915843610512988279987

Table 3. Successive approximations to freq(1) using periodic points up to period n

n

5  0.23197531592304682578341152207518262438
6  0.23145978794196133643466232097577697555
7  0.23150435225320446593755544220404095829
8  0.23150108449559544202968446721419088614
9  0.23150134604955240614907489089593967961
10  0.2315013239226018884286778845661499860
11  0.2315013257492966889337542141808042650
12  0.2315013256017684253229890778051297912

Table 4. Successive approximations to freq(2) using periodic points up to period n

Proof. Taking f = ∑_{j=0}^{2} jχ_{I_{j}}, Birkhoff’s ergodic theorem gives the empirical estimate

lim_{n→∞} \frac{1}{n} (a_{1} + … + a_{n}) = ∫ f dμ = freq(1) + 2freq(2) = 0.767421696,

by the empirical results of Theorem 2. The rigorous error bound is given by summing the three error bounds in Theorem 2.

7. Rate of Correlation Decay

In this final section we compute empirically the rate of correlation decay of analytic functions (observables) for the Bolyai-Rényi dynamical system (T, μ).

For analytic functions f, g, define the n^{th} correlation function

c_{n}(f, g) = ∫ f · T^{n}g dμ − ∫ f dμ ∫ g dμ.

Lemma 3. The exponential rate of correlation decay

sup \{lim sup_{n→∞} \sqrt[n]{c_{n}(f, g)} : f, g ∈ A\}

is given by the second-largest (in modulus) eigenvalue α of the Perron-Frobenius operator L.

Proof. First note that

\begin{aligned}∫ f · T^{n}g dμ & = ∫ f(T^{n}x)g(x)ϱ(x) dx \\ & = ∫ f(x)L^{n}(gϱ)(x) dx, \end{aligned}

by the change of variable formula, where ϱ = Lϱ denotes the density function dμ/dλ.

Writing g = c + (g − c), where c = ∫ g dμ, we have

L^{n}(gϱ) = L^{n}(cϱ) + L^{n} ((g − c)ϱ)
= cϱ + L^{n} ((g − c)ϱ)
= cϱ + ϱ ∫ (g − c) dμ + O(α^{n})
= ϱ ∫ g dμ + O(α^{n}),

with respect to the uniform norm (see Ruelle [Ru2]). Therefore

\begin{aligned}∫ f(x)L^{n}(gϱ)(x) dx & = ∫ f(x) (ϱ(x) ∫ g dμ + O(α^{n})) dx \\ & = ∫ f dμ ∫ g dμ + O(α^{n}), \end{aligned}

as required.

The fact that L is nuclear (set t = 0 in Proposition 3) means its eigenvalues (with algebraic multiplicities) are precisely the reciprocals of the zeros (again with algebraic multiplicities) of the Fredholm determinant d(z) = F (z, 0) = 1 + ∑_{m=1}^{∞} C_{m}(0)z^{m}, see [Gr]. In particular, the modulus α_{n} of the second smallest zero of the polynomial truncation d_{n}(z) = 1 + ∑_{m=1}^{n} C_{m}(0)z^{m} converges to the rate of correlation decay α.

The α_{n} are listed in Table 5, for n = 1, …, 12.

Theorem 3. The rate of correlation decay of the Bolyai-Rényi expansion, with respect to the absolutely continuous invariant measure μ, is 0.40582 ± 0.000076.

Moreover, this rate of correlation decay is given empirically as 0.40582.

Proof. We will use Rouché’s theorem to compare the zeros of the Fredholm determinant d(z) to those of its twelth order approximation d_{12}(z). The first twelve coefficients C_{1}(0), …, C_{12}(0) are given in Table 6.

More precisely, for suitable choices of circular contour Γ_{i} we will show that

|d(z) − d_{12}(z)| < |d_{12}(z)| for all z ∈ Γ_{i},  (7.1)

and deduce that d, d_{12} have the same number of zeros in the disc enclosed by Γ_{i}.

The term |d(z) − d_{12}(z)| = ∑_{m=13}^{∞} C_{m}(0)z^{m} is estimated by means of the bound (4.3).

The three leading zeros (i.e. zeros of minimum modulus) of d_{12} are

\begin{aligned}z_{1} & = 0.99999999999559060219576663208556880538719, \\ z_{2} & = 2.46409306770808208005653362549270757380082, \\ z_{3} & = 5.20333607178584420947630332332064588062949. \end{aligned}

Taking Γ_{1} to be the circle centred at \frac{1}{2}(z_{1} + z_{2}), and of radius 3/4, say, we easily check that (7.1) holds and deduce that the Fredholm determinant d has exactly two zeros in the disc A_{1} bounded by Γ_{1}.

Now one of these zeros is clearly the value 1 (the reciprocal of the dominant eigenvalue for the Perron-Frobenius operator), so the other zero of d in the disc A_{1}, let us call it ζ_{2}, will determine the rate of correlation decay α = 1/|ζ_{2}|.

Taking Γ_{2} to be the circle centred at z_{2}, of radius 0.00046, the bound (4.3) gives that max_{z∈Γ_{2}} |d(z)−d_{12}(z)| < 0.0000815564. But min_{z∈Γ_{2}} |d_{12}(z)| > 0.0000824847, so that (7.1) again holds. Therefore d has exactly one zero in the disc A_{2} bounded by Γ_{2}. Since A_{2} ⊂ A_{1} then this zero must be ζ_{2}. Thus we have |ζ_{2} − z_{2}| < 0.00046.

Therefore

|\frac{1}{ζ_{2}} − \frac{1}{z_{2}} | = \frac{|ζ_{2} − z_{2}|}{z_{2}(z_{2} − 0.00046)} ≤ 0.0000757748,

which implies that |α − α_{12}| < 0.000076, as required.

For the empirical estimate on the rate of correlation decay we compare the values of α_{11} and α_{12} in Table 5.

n
5  0.43565314532422384499908815386394606459
6  0.39710261645398773605165411752521458435
7  0.40692087304013760480215803501542286844
8  0.40564938929243344663394558917323398927
9  0.40586126580655750137047565327375013481
10  0.4058232124663283507047545731401967112
11  0.4058300461512405617270535004484480744
12  0.4058288272894360932942766784291223501

Table 5. Successive approximations to the rate of correlation decay

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[Bol] W. Bolyai, Tentamen iuventutem studiosam in elementa matheseos purae elementaris ac sublimioris nethodo intuitiva evidentiaque huic propria introducendi, ed. sec., Vol. 1, Budapest, 1897.

[BDK] W. Bosma, K. Dajani, and C. Kraaikamp, Entropy and counting correct digits, preprint

\begin{matrix}m & C_{m}(0) \\ 1 & -1.7805469288332912726151680670795885804214570052556384781875 \\ 2 & 0.968150202409978866190931425171238046258873082241796047534 \\ 3 & -0.200449499881595422668099261082517627176267155823210543450 \\ 4 & 0.011747548689539614515696836102251495253538538089366139039 \\ 5 & 0.001320729616885514423440075737825684210946681816268338684 \\ 6 & -0.00023845939808201314790525094806777333962592582638710399 \\ 7 & 0.000017377487824315663577513619371577982425956328823478337 \\ 8 & -0.000001034099171687213435387495713380529939649979420805256 \\ 9 & 0.000000068690346747371206658806483281475344799644156631040 \\ 10 & -0.00000000502354667531727959402600313654340414663735337001 \\ 11 & 0.00000000036586403391418295704578835805521809268377948334 \\ 12 & -0.0000000000264863166246360636702410679182409538505237410 \end{matrix}

Table 6. First 12 power series coefficients C_{m}(0) of the Fredholm determinant d(z) = F (z, 0)

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[Ru1] D. Ruelle, Zeta functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), 231–242.

[Ru2] D. Ruelle, Thermodynamic Formalism, Wiley, New York, 1978.

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O. Jenkinson, UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France.

   E-mail address: omj@iml.univ-mrs.fr  ,  omj@maths.warwick.ac.uk

M. Pollicott, Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

   E-mail address: mp@ma.man.ac.uk